Browne — On an Integral Equation proposed by Abel. 

 We can prove in a similar manner that the integrals 



x« [U(a)] y 1 "\f/(y)dy 



Zr{x) ~ 2ni) D 1-A M « 



have a meaning when r > 0. Hence the formula 



da 



* ( * ) = 2^ 



Ca ) () ' jA ' a ^^ 



dy 



da 



d1-\/j,«-\ t«K{t)dt 

 gives a solution of the equation 



<!> (x) - \<j> {fix) = ^ (x) + T K(t) <j> (tx) dt. 

 We can easily see, as in the case /u = 0, that a solution of the equation 



\^G(t)f(tx)dt=g(.r), (0(1) + 0), 

 ^ given by . x^\\j-^g(y)dy 



f{ \ - 1 d 

 T T } ~ 2nidx 



da. 



■D(l + a) J' t«G(t)dt 



8. The solution of the more general equation 



j 1 G(x,t)f(tx)dt = g(x) 



can now be readily found by successive approximation. Setting 



*(*)- J* /(*)**, 

 and integrating by parts as before, we find an equation of the form 



<j> (x) - h (x) $ (/xx) = if, (x) + I K{x, t) <p (tx) dt. 



Let us first take the case /x = 0, in which we have the equation 



4> (x) = i// (x) + j ' K (x, t) $ {tx) dt. 



Setting K(0, t) = K{t), and K(x, t) - K(Q, t) = r, {x, t), we have 



(j)(x) 



\p{x) + \ t) (x, t) <j>(tx) dt + K{t) (j> (tx) dt, 



from which we deduce <j> (x) = <p (x) + S<p (x) , where 



O («) 



23« v(a) , If 



X a 



\" y- 1 -°-4'(y)dy 



and 



S<p(x) = 



ci-Vt*K(t)dt ' Ziri ^ l-\t*K(t)dt 



1 f x«\ a y-i-*\\\(y,t)${ty) dt\da. 



fZa, 



2iri] £, 



to- K(t) dt 



[11*] 



77 



